Invariants
| Level: | $28$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $4^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $2$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-7$) |
Other labels
| Cummins and Pauli (CP) label: | 4F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.12.0.16 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}5&12\\0&19\end{bmatrix}$, $\begin{bmatrix}14&9\\27&26\end{bmatrix}$, $\begin{bmatrix}15&6\\6&27\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 28-isogeny field degree: | $16$ |
| Cyclic 28-torsion field degree: | $192$ |
| Full 28-torsion field degree: | $16128$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 44 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{x^{12}(x^{2}-30xy-27y^{2})^{3}(17x^{2}-6xy+45y^{2})^{3}}{x^{12}(x-y)^{4}(2x^{2}+3xy+9y^{2})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 28.24.0.b.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0.d.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0.g.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0.h.1 | $28$ | $2$ | $2$ | $0$ |
| 28.96.6.p.1 | $28$ | $8$ | $8$ | $6$ |
| 28.252.15.v.1 | $28$ | $21$ | $21$ | $15$ |
| 28.336.21.cr.1 | $28$ | $28$ | $28$ | $21$ |
| 56.24.0.r.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.z.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.cj.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.cr.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.db.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.dd.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.dx.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.ed.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.ej.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.el.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.ep.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.ev.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.1.bh.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.bn.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.br.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.bt.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.bx.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.cd.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.cx.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.cz.1 | $56$ | $2$ | $2$ | $1$ |
| 84.24.0.j.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.l.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.z.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.bc.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.1.fh.1 | $84$ | $3$ | $3$ | $1$ |
| 84.48.2.m.1 | $84$ | $4$ | $4$ | $2$ |
| 140.24.0.l.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.n.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.v.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.y.1 | $140$ | $2$ | $2$ | $0$ |
| 140.60.4.cb.1 | $140$ | $5$ | $5$ | $4$ |
| 140.72.3.er.1 | $140$ | $6$ | $6$ | $3$ |
| 140.120.7.hl.1 | $140$ | $10$ | $10$ | $7$ |
| 168.24.0.em.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.fa.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.hr.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.ij.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.jf.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.jh.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.kr.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.kx.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mx.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mz.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nd.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nj.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.1.fh.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.fn.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.fr.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.ft.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.jd.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.jj.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.kt.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.kv.1 | $168$ | $2$ | $2$ | $1$ |
| 252.324.22.hr.1 | $252$ | $27$ | $27$ | $22$ |
| 280.24.0.es.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fg.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hp.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ih.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jd.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jf.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.kp.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.kv.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.mv.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.mx.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.nb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.nh.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.ej.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ep.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.et.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ev.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.iz.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.jf.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.kp.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.kr.1 | $280$ | $2$ | $2$ | $1$ |
| 308.24.0.n.1 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0.p.1 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0.t.1 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0.x.1 | $308$ | $2$ | $2$ | $0$ |
| 308.144.10.h.1 | $308$ | $12$ | $12$ | $10$ |